A billable expense is basically an expense that you have incurred on behalf of your client/customer for performing work, services, supplies.
Billable expenses are reimbursable from your customer simply by billing them the incurred expenses when you send them an invoice.
Some examples of these may be travel expenses, advertising costs or costs for materials.
It is important to ensure that you and your client have agreed on the billable expenses and that your client is aware that they will be billed for these expenses incurred.
Also you want to make sure to keep those receipts and any supporting evidence such as emails/confirmations of the incurred expenses.